Before we begin the proof of Theorem REF  proper, we review the various formulations of the Torelli theorem for K3 surfaces.
Suppose \(M\)  is endowed with a complex structure \(J\)  (so that by [1]} it becomes a K3 surface). It is known that the automorphism group \(\operatorname{Aut}(M, J)\)  acts faithfully on \(\mathbf {\Lambda }\)  and so can be considered as a subgroup of \(\operatorname{O}^+(\mathbf {\Lambda })\) . The complex structure gives rise to
a Hodge decomposition of \(\mathbf {\Lambda }\)  with \(\dim H^{2,0}(M,J)=1\) , and basic Hodge theory tells us that this complex line is isotropic
for the complex-bilinear extension of the intersection pairing and positive definite for the hermitian extension. This implies
that \(H^{2,0}(M,J)\)  maps under the projection \(\mathbf {\Lambda }_{\mathbb {C}}\rightarrow \mathbf {\Lambda }_{\mathbb {R}}\)  (taking the real part) onto a positive-definite 2-plane \(\Pi _J\subset \mathbf {\Lambda }_{\mathbb {R}}\) . This 2-plane \(\Pi _J\)  inherits a natural orientation from \(H^{2,0}(M,J)\) , and as such it is easily seen to be a complete invariant of the Hodge structure.
